Newton's Bucket: Fluid in a Spinning Tank скачать в хорошем качестве

Newton's Bucket: Fluid in a Spinning Tank

About the Video: In Newton’s famous bucket experiment, water in a spinning container doesn’t stay level — instead, its surface rises at the edges and sinks in the middle, forming a paraboloid. This shape isn’t a coincidence: it’s the natural ‘flat’ surface in a rotating frame, just as a sphere is ‘flat’ under a planet’s radial gravity. In this video, we explore why the paraboloid appears, what it tells us about rotation, and how it connects to the idea of gravitational equipotential surfaces. ------------------------------------------------------------------------------------------------------------------------------------------------------------------ The Origins of Newton’s Bucket: Newton’s Bucket gets its name from Isaac Newton’s famous experiment, described in his Principia. When a bucket of water is spun, the surface curves into a paraboloid – when the water and bucket rotate together. Newton used this simple setup to argue that the effect comes from motion relative to absolute space, not just relative motion between objects. This idea sparked centuries of debate about the nature of space and motion. ------------------------------------------------------------------------------------------------------------------------------------------------------------------ The Calculus: i) Momentum balance in rotating frame (per unit mass): 0 = -∇(p/ρ) - ∇(gz - 1/2 * Omega^2 * r^2) (Interpretation: pressure gradient balances "effective potential" made of gravity (gz) and centrifugal ( -1/2 Omega^2 r^2 ).) ii) Integrate the gradient: p/ρ + gz - (Omega^2 / 2) * r^2 = C (C = constant) iii) Apply free-surface condition (p = constant at the surface): gz - (Omega^2 / 2) * r^2 = C' (absorb p/ρ into C') Solve for the surface shape: z(r) = z0 + (Omega^2 / (2g)) * r^2 iv) Rim–center height difference (at r = R): Delta_h = z(R) - z(0) = (Omega^2 * R^2) / (2g) v) Volume of the raised paraboloid (bulge relative to flat z = 0): Treat z(r) as a surface of revolution, use cylindrical shells: V = pi * ∫[0→R] z(r) * r dr = pi * ∫[0→R] (Omega^2 / (2g) * r^2) * r dr = (pi * Omega^2 / (2g)) * ∫[0→R] r^3 dr = (pi * Omega^2 / (2g)) * (R^4 / 4) = (pi * Omega^2 * R^4) / (8g) ~~~ Calculus in Typed PDF Form ~~~ https://drive.google.com/file/d/1AEPk... https://drive.google.com/file/d/1cVEh... ------------------------------------------------------------------------------------------------------------------------------------------------------------------ Additional Resources:    • Spinning Water Shape Derivation   https://www.usna.edu/Users/physics/mu...